In this appendix, we consider a C1−functionϕfor a domain Ω0 into itself such thatϕis a local diffeo-morphism andϕ|∂Ω= id. We would like to obtain thatϕis in fact a global diffeomorphism from Ω0into itself. This extension needs topological arguments contained in the article [41] of Meisters and Olech.
Theorem 1 of [41] applied to the ball ofRN writes as follows.
Theorem A.5. (Meisters-Olech, 1963)
Let BR be the open ball of center 0 and radius R > 0 of RN and let BR the closed ball. Let f be a continuous mapping of BR into itself which is locally one-to-one on BR\Z, where Z∩BR is discrete and Z does not cover the whole boundary ∂BR. If f is one-to-one from ∂BR into itself, then f is an homeomorphism of BR onto itself.
In fact, the original result of [41] includes different domains than the balls. Nevertheless, if we consider any smooth domain, then it has to be diffeomorphic to a ball (typically, annulus are excluded).
To consider more general domains, we assume that f is the identity at the boundary.
Theorem A.6. Let Ω⊂RN be a bounded open domain. Letf be a continuous mapping of Ωinto itself which is locally one-to-one on Ω\Z, whereZ is a finite set. Assume moreover thatf is the identity on
∂Ω. Then,f is an homeomorphism ofΩonto itself.
Proof: ForR large enough, Ω is included inside the ballBR. We extend continuouslyf to a function f˜by setting ˜f = id on BR\Ω. Notice that f maps Ω into itself and is locally one-to-one at all the points of the boundary, except maybe at a finite number of them. This yields that ˜f is locally one-to-one at all these points since the extension maps the outside of Ω into itself. We apply Theorem A.5 to ˜f and obtain that ˜f is an homeomorphism ofBR. Since it is the identity outside Ω,f = ˜f|Ωmust be an
homeomorphism of Ω.
If we consider f of class Ck and Df its jacobian matrix, then we may check the local one-to-one property by assuming that det(Df) only vanishes at a finite number of points. More importantly, if det(Df) never vanishes, thenf is a diffeomorphism.
Corollary A.7. Let k≥1, let Ω⊂RN be a bounded open domain of classCk and let f ∈ Ck(Ω,Ω). As-sume thatdet(Df)does not vanish onΩand thatf is the identity on∂Ω. Then,f is aCk−diffeomorphism of Ωonto itself.
We could also be interested in the following other consequence.
Corollary A.8. Let k ≥ 1, let Ω ⊂ RN be a bounded open domain of class Ck and let f be a Ck−diffeomorphism from Ω onto itself. Assume that f is the identity on ∂Ω. Then, for all ε > 0, there exists η >0 such that, for all functions g∈ Ck(Ω,Ω)with g∂Ω= idand kf−gkCk≤η,g is also a Ck−diffeomorphism ofΩ onto itself andkf−1−g−1kCk≤ε.
Proof: We simply notice that Ω is compact and so |det(Df)| ≥ α > 0 for some uniform positive α.
Thus, for g which is C1−close to f, Dg is still invertible everywhere andg is aCk−diffeomorphism due
to Corollary A.7.
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